And I have another topic suggested by John Golden, author of Math Hombre. It’s one of the basic bits of mathematics, and so is hard to think about.
Edward Brisse assembled a list of 2,001 things to call a “center” of a triangle. I’d have run out around three. We don’t need most of them. I mention them because the list speaks of how interesting we find triangles. Nobody’s got two thousand thoughts about enneadecagons (19-sided figures).
As always with mathematics it’s hard to say whether triangles are all that interesting or whether we humans are obsessed. They’ve got great publicity. The Pythagorean Theorem may be the only bit of interesting mathematics an average person can be assumed to recognize. The kinds of triangles — acute, obtuse, right, equilateral, isosceles, scalene — are fit questions for trivia games. An ordinary mathematics education can end in trigonometry. This ends up being about circles, but we learn it through triangles. The art and science of determining where a thing is we call “triangulation”.
But triangles do seem to stand out. They’re the simplest polygon, only three vertices and three edges. So we can slice any other polygon into triangles. Any triangle can tile the plane. Even quadrilaterals may need reflections of themselves. One of the first geometry facts we learn is the interior angles of a triangle add up to two right angles. And one of the first geometry facts we learn, discovering there are non-Euclidean geometries, is that they don’t have to.
Triangles have to be convex, that is, they don’t have any divots. This property sounds boring. But it’s a good boring; it makes other work easier. It tells us that the length of any two sides of a triangle add together to something longer than the third side. And that’s a powerful idea.
There are many ways to define “distance”. Mathematicians have tried to find the most abstract version of the concept. This inequality is one of the few pieces that every definition of “distance” must respect. This idea of distance leaps out of shapes drawn on paper. Last week I mentioned a triangle inequality, in discussing functions and . We can define operators that describe a distance between functions. And the distances between trios of functions behave like the distances between points on the triangle. Thus does geometry sneak in to abstract concepts like “piecewise continuous functions”.
And they serve in curious blends of the abstract and the concrete. For example, numerical solutions to partial differential equations. A partial differential equation is one where we want to know a function of two or more variables, and only have information about how the function changes as those variables change. These turn up all the time in any study of things in bulk. Heat flowing through space. Waves passing through fluids. Fluids running through channels. So any classical physics problem that isn’t, like, balls bouncing against each other or planets orbiting stars. We can solve these if they’re linear. Linear here is a term of art meaning “easy”. I kid; “linear” means more like “manageable”. All the good problems are nonlinear and we can exactly solve about two of them.
So, numerical solutions. We make approximations by putting down a mesh on the differential equation’s domain. And then, using several graduate-level courses’ worth of tricks, approximating the equation we want with one that we can solve here. That mesh, though? … It can be many things. One powerful technique is “finite elements”. An element is a small piece of space. Guess what the default shape for these elements are. There are times, and reasons, to use other shapes as elements. You learn those once you have the hang of triangles. (Dividing the space of your variables up into elements lets you look for an approximate solution using tools easier to manage than you’d have without. This is a bit like looking for one’s keys over where the light is better. But we can find something that’s as close as we need to our keys.)
If we need finite elements for, oh, three dimensions of space, or four, then triangles fail us. We can’t fill a volume with two-dimensional shapes like triangles. But the triangle has its analog. The tetrahedron, in some sense four triangles joined together, has all the virtues of the triangle for three dimensions. We can look for a similar shape in four and five and more dimensions. If we’re looking for the thing most like an equilateral triangle, we’re looking for a “simplex”.
These simplexes, or these elements, sprawl out across the domain we want to solve problems for. They look uncannily like the triangles surveyors draw across the chart of a territory, as they show us where things are.
Next week I hope to cover the letter ‘I’ as I near the end of ‘Mathematics’ and consider what to do about ‘A To Z’. This week’s essay, and all the essays for the Little Mathematics A-to-Z, should be at this link. And all of this year’s essays, and all the A-to-Z essays from past years, should be at this link. Thank you once more for reading.